Promise problems solved by quantum and classical finite automata

TL;DR

This paper investigates promise problems within classical, quantum, and semi-quantum finite automata, revealing their recognition and solvability capabilities and demonstrating quantum automata's advantages over classical models.

Contribution

It introduces acceptance modes for promise problems and establishes new complexity results showing quantum automata's superior recognition and solving power compared to classical automata.

Findings

Quantum automata can recognize certain promise problems that classical automata cannot.

Quantum automata can solve promise problems with fewer states and lower error rates.

Classical automata require significantly more states to solve some promise problems.

Abstract

The concept of promise problems was introduced and started to be systematically explored by Even, Selman, Yacobi, Goldreich, and other scholars. It has been argued that promise problems should be seen as partial decision problems and as such that they are more fundamental than decision problems and formal languages that used to be considered as the basic ones for complexity theory. The main purpose of this paper is to explore the promise problems accepted by classical, quantum and also semi-quantum finite automata. More specifically, we first introduce two acceptance modes of promise problems, recognizability and solvability, and explore their basic properties. Afterwards, we show several results concerning descriptional complexity on promise problems. In particular, we prove: (1) there is a promise problem that can be recognized exactly by measure-once one-way quantum finite automata (MO-1QFA), but no deterministic finite automata (DFA) can recognize it; (2) there is a promise problem that can be solved with error probability $\epsilon\leq 1/3$ by one-way finite automaton with quantum and classical states (1QCFA), but no one-way probability finite automaton (PFA) can solve it with error probability $\epsilon\leq 1/3$; and especially, (3) there are promise problems $A(p)$ with prime $p$ that can be solved {\em with any error probability} by MO-1QFA with only two quantum basis states, but they can not be solved exactly by any MO-1QFA with two quantum basis states; in contrast, the minimal PFA solving $A(p)$ with any error probability (usually smaller than $1/2$) has $p$ states. Finally, we mention a number of problems related to promise for further study.